No, not directly, at least not yet. There are multiple supplemental databases: minor planets, comets, stars, and deep sky objects. There have been changes to all of them, usually meaning more data is required for ST4, so any import is going to end up being limited in some way. There isn't a need to import minor planets and comets. Most of the changes to supplemental stellar database involved novae and supernova and I don't recommend importing those. Single stars are easier. As for deep sky objects, the main difference is that ST4 has an alternative ID field, which will end up being blank.

The one way I can think of to currently import objects from ST3 is to export them as part of an observing list. When you import the list into ST4 it will add the objects to the Supplemental database. The hard part there is creating the list. I can't see a way to use the Database Power Search to do this. Your best bet would be to use the Browse Tab of the Designation Search tool. You would select, say, Supplemental Deep Sky as the Reference database. Create a new observing list, and then select each object in turn and add it to the new list. This isn't as bad as it sounds, unless you have many hundreds of objects.

I said that there was no way to do this *yet*. It is not a priority for a simple reason: I never expected anyone to add so many supplemental database objects. It is way outside of normal usage. As such it is not a high priority, particularly right now when I have so many other much more pressing things to do.